RealClosure TheField

reclos.spad line 866 [edit on github]

• TheField: Join(OrderedRing, Field, RealConstant)

This domain implements the real closure of an ordered field.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (%, Fraction Integer) -> %

from RightModule Fraction Integer

*: (%, Integer) -> %

from RightModule Integer

*: (%, TheField) -> %

from RightModule TheField

*: (Fraction Integer, %) -> %

from LeftModule Fraction Integer

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (TheField, %) -> %

from LeftModule TheField

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, %) -> %

from Field

<=: (%, %) -> Boolean

from PartialOrder

<: (%, %) -> Boolean

from PartialOrder

=: (%, %) -> Boolean

from BasicType

>=: (%, %) -> Boolean

from PartialOrder

>: (%, %) -> Boolean

from PartialOrder

^: (%, Fraction Integer) -> %

from RadicalCategory

^: (%, Integer) -> %

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

abs: % -> %

from OrderedRing

algebraicOf: (RightOpenIntervalRootCharacterization(%, SparseUnivariatePolynomial %), OutputForm) -> %

algebraicOf(char) is the external number

allRootsOf: Polynomial % -> List %

from RealClosedField

allRootsOf: Polynomial Fraction Integer -> List %

from RealClosedField

allRootsOf: Polynomial Integer -> List %

from RealClosedField

allRootsOf: SparseUnivariatePolynomial % -> List %

from RealClosedField

allRootsOf: SparseUnivariatePolynomial Fraction Integer -> List %

from RealClosedField

allRootsOf: SparseUnivariatePolynomial Integer -> List %

from RealClosedField

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

approximate: (%, %) -> Fraction Integer

from RealClosedField

associates?: (%, %) -> Boolean

from EntireRing

associator: (%, %, %) -> %

from NonAssociativeRng

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

coerce: % -> %

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Fraction Integer -> %

from CoercibleFrom Fraction Integer

coerce: Integer -> %

from CoercibleFrom Integer

coerce: TheField -> %

from CoercibleFrom TheField

commutator: (%, %) -> %

from NonAssociativeRng

divide: (%, %) -> Record(quotient: %, remainder: %)

from EuclideanDomain

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

expressIdealMember: (List %, %) -> Union(List %, failed)

from PrincipalIdealDomain

exquo: (%, %) -> Union(%, failed)

from EntireRing

extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %)

from EuclideanDomain

extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed)

from EuclideanDomain

factor: % -> Factored %

from UniqueFactorizationDomain

gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %

from GcdDomain

inv: % -> %

from DivisionRing

latex: % -> String

from SetCategory

lcm: (%, %) -> %

from GcdDomain

lcm: List % -> %

from GcdDomain

lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %)

from LeftOreRing

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

mainCharacterization: % -> Union(RightOpenIntervalRootCharacterization(%, SparseUnivariatePolynomial %), failed)

mainCharacterization(x) is the main algebraic quantity of x (SEG)

mainDefiningPolynomial: % -> Union(SparseUnivariatePolynomial %, failed)

from RealClosedField

mainForm: % -> Union(OutputForm, failed)

from RealClosedField

mainValue: % -> Union(SparseUnivariatePolynomial %, failed)

from RealClosedField

max: (%, %) -> %

from OrderedSet

min: (%, %) -> %

from OrderedSet

multiEuclidean: (List %, %) -> Union(List %, failed)

from EuclideanDomain

negative?: % -> Boolean

from OrderedRing

nthRoot: (%, Integer) -> %

from RadicalCategory

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra Integer

positive?: % -> Boolean

from OrderedRing

prime?: % -> Boolean

from UniqueFactorizationDomain

principalIdeal: List % -> Record(coef: List %, generator: %)

from PrincipalIdealDomain

quo: (%, %) -> %

from EuclideanDomain

recip: % -> Union(%, failed)

from MagmaWithUnit

relativeApprox: (%, %) -> Fraction Integer

relativeApprox(n, p) gives a relative approximation of n that has precision p

rem: (%, %) -> %

from EuclideanDomain

rename!: (%, OutputForm) -> %

from RealClosedField

rename: (%, OutputForm) -> %

from RealClosedField

retract: % -> Fraction Integer

from RetractableTo Fraction Integer

retract: % -> Integer

from RetractableTo Integer

retract: % -> TheField

from RetractableTo TheField

retractIfCan: % -> Union(Fraction Integer, failed)

from RetractableTo Fraction Integer

retractIfCan: % -> Union(Integer, failed)

from RetractableTo Integer

retractIfCan: % -> Union(TheField, failed)

from RetractableTo TheField

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

rootOf: (SparseUnivariatePolynomial %, PositiveInteger) -> Union(%, failed)

from RealClosedField

rootOf: (SparseUnivariatePolynomial %, PositiveInteger, OutputForm) -> Union(%, failed)

from RealClosedField

sample: %

from AbelianMonoid

sign: % -> Integer

from OrderedRing

sizeLess?: (%, %) -> Boolean

from EuclideanDomain

smaller?: (%, %) -> Boolean

from Comparable

sqrt: % -> %

from RealClosedField

sqrt: (%, PositiveInteger) -> %

from RealClosedField

sqrt: Fraction Integer -> %

from RealClosedField

sqrt: Integer -> %

from RealClosedField

squareFree: % -> Factored %

from UniqueFactorizationDomain

squareFreePart: % -> %

from UniqueFactorizationDomain

subtractIfCan: (%, %) -> Union(%, failed)

from CancellationAbelianMonoid

unit?: % -> Boolean

from EntireRing

unitCanonical: % -> %

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %)

from EntireRing

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

Algebra Integer

Algebra TheField

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

BiModule(Integer, Integer)

BiModule(TheField, TheField)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicZero

CoercibleFrom Fraction Integer

CoercibleFrom Integer

CoercibleFrom TheField

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable

DivisionRing

EntireRing

EuclideanDomain

Field

FullyRetractableTo Fraction Integer

FullyRetractableTo TheField

GcdDomain

IntegralDomain

LeftModule %

LeftModule Fraction Integer

LeftModule Integer

LeftModule TheField

LeftOreRing

Magma

MagmaWithUnit

Module %

Module Fraction Integer

Module Integer

Module TheField

Monoid

NonAssociativeAlgebra %

NonAssociativeAlgebra Fraction Integer

NonAssociativeAlgebra Integer

NonAssociativeAlgebra TheField

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

OrderedAbelianGroup

OrderedAbelianMonoid

OrderedAbelianSemiGroup

OrderedCancellationAbelianMonoid

OrderedRing

OrderedSet

PartialOrder

PrincipalIdealDomain

RadicalCategory

RealClosedField

RetractableTo Fraction Integer

RetractableTo Integer

RetractableTo TheField

RightModule %

RightModule Fraction Integer

RightModule Integer

RightModule TheField

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown